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TR 537 Approved 2003-05 HANDBOOK FOR CALCULATION OF MEASUREMENT UNCERTAINTY IN ENVIRONMENTAL LABORATORIES Bertil Magnusson Teemu Näykki Håvard Hovind Mikael Krysell Published by Nordtest Tekniikantie 12 FIN–02150 Espoo Finland Phone: + 358 9 455 4600 E-mail: [email protected] Fax: + 358 9 455 4272 Internet: www.nordtest.org NT TECHN REPORT 537 Approved 2003-05 Authors: Bertil Magnusson1 Teemu Näykki2 Håvard Hovind3 Mikael Krysell4 NORDTEST project number: 1589-02 Institution: SP, Sweden, 2) SYKE, Finland, 3) NIVA, Norway, 4) Eurofins A/S, Denmark 1) Title (English): Title (Original): Handbook for Calculation of Measurement Uncertainty in Environmental Laboratories Abstract: This handbook is written for environmental testing laboratories in the Nordic countries, in order to give support to the implementation of the concept of measurement uncertainty for their routine measurements. The aim is to provide a practical, understandable and common way of measurement uncertainty calculations, mainly based on already existing quality control and validation data, according to the European accreditation guideline /12/, the Eurolab Technical Report No. 1 /3/ and the ISO/DTS 21748 Guide /8/. Nordtest has supported this project economically in order to promote and enhance harmonisation between laboratories on the Nordic market. Practical examples, taken directly from the everyday world of environmental laboratories, are presented and explained. However, the approach is very general and should be applicable to most testing laboratories in the chemical field. The handbook covers all steps in the analytical chain from the arrival of the sample in the laboratory until the data has been reported. It is important to notice that vital parts of the total measurement uncertainty are not included, e.g. sampling, sample transportation and possible gross errors during data storage/retrieval. The recommendations in this document are primarily for guidance. It is recognised that while the recommendations presented do form a valid approach to the evaluation of measurement uncertainty for many purposes, other suitable approaches may also be adopted – see references in Section 9. Especially the EURACHEM/CITAC-Guide /2/ is useful in cases where sufficient previous data is not available, and therefore the mathematical analytical approach according to GUM /1/ with all different steps is to be used. Basic knowledge in the use of quality control and statistics is required. In order to make it possible for the reader to follow the calculations, some raw data is given in appendices. Technical Group: Expert Group Quality and Metrology ISSN: 0283-7234 Language: English Class (UDC): 620.1 Key words: Distributed by: NORDTEST Tekniikantie 12 FIN-02150 ESPOO Finland Publication code: Pages: 41 handbook, laboratories, environmental, testing, measurements, uncertainty, chemical Report Internet address: http://www.nordtest.org/register/techn/tlibrary/tec537.pdf Handbook for Calculation of Measurement Uncertainty in Environmental Laboratories Version 1.2 June 2003 Nordtest project 1589-02 Project participants Bertil Magnusson, SP, Sweden Teemu Näykki, SYKE, Finland Håvard Hovind, NIVA, Norway Mikael Krysell, Eurofins A/S, Denmark Drawings by Petter Wang, NIVA, Norway Valuable comments on the contents have been provided by: Rolf Flykt, Sweden Irma Mäkinen, Finland Ulla O. Lund, Denmark Steve Ellison, UK Contents: 1 2 DEFINITIONS AND ABBREVIATIONS ...................................................... 1 INTRODUCTION............................................................................................ 3 2.1 2.2 2.3 3 CALCULATION OF EXPANDED UNCERTAINTY, U - OVERVIEW ...... 6 3.1 3.2 3.3 4 CERTIFIED REFERENCE MATERIAL......................................................................... 15 INTERLABORATORY COMPARISONS ........................................................................ 17 RECOVERY ............................................................................................................. 18 REPRODUCIBILITY BETWEEN LABORATORIES, SR ........................... 19 6.1 6.2 7 CUSTOMER DEMANDS ............................................................................................ 10 CONTROL SAMPLE COVERING THE WHOLE ANALYTICAL PROCESS.......................... 10 CONTROL SAMPLE FOR DIFFERENT MATRICES AND CONCENTRATION LEVELS ........ 11 UNSTABLE CONTROL SAMPLES............................................................................... 12 METHOD AND LABORATORY BIAS – u(BIAS) ................................... 15 5.1 5.2 5.3 6 CUSTOMER NEEDS .................................................................................................... 7 FLOW SCHEME FOR UNCERTAINTY CALCULATIONS .................................................. 7 SUMMARY TABLE FOR UNCERTAINTY CALCULATIONS ............................................. 9 REPRODUCIBILITY WITHIN-LABORATORY - u(R W) .......................... 10 4.1 4.2 4.3 4.4 5 SCOPE AND FIELD OF APPLICATION .......................................................................... 3 COMMENT TO CUSTOMERS ....................................................................................... 3 ABOUT MEASUREMENT UNCERTAINTY.................................................................... 4 DATA GIVEN IN STANDARD METHOD ...................................................................... 19 DATA FROM INTERLABORATORY COMPARISONS .................................................... 19 EXAMPLES................................................................................................... 21 7.1 7.2 7.3 7.4 AMMONIUM IN WATER ........................................................................................... 21 BOD IN WASTEWATER ........................................................................................... 21 PCB IN SEDIMENT .................................................................................................. 25 CONCENTRATION RANGES ...................................................................................... 28 8 REPORTING UNCERTAINTY.................................................................... 30 9 REFERENCES............................................................................................... 32 10 APPENDICES................................................................................................ 33 Appendix 1: Empty flow scheme for calculations.................................................. 33 Appendix 2: Empty summary table ........................................................................ 34 Appendix 3: Error model used in this handbook .................................................... 35 Appendix 4: Uncertainty of bias for NH4-N in section 3.2..................................... 36 Appendix 5: Raw data for NH4-N in section 4.3 .................................................... 37 Appendix 6: Raw data for oxygen in Section 4.4 ................................................... 39 Appendix 7: Raw data for BOD in example 6.1..................................................... 40 Appendix 8: Estimation of standard deviation from range ..................................... 41 1 Definitions and abbreviations s An estimate of the population standard deviation σ from a limited number (n) of observations (xi) x Mean value u(x) Individual standard uncertainty component (GUM, /1/). uc Combined standard uncertainty (GUM, /1/) U Expanded combined uncertainty close to 95 % confidence interval r Repeatability limit – performance measure for a test method or a defined procedure when the test results are obtained under repeatability conditions. Repeatability conditions: Conditions where independent test results are obtained with the same method on identical test items in the same laboratory by the same operator using the same equipment within short intervals of time. Repeatability (precision under repeatability conditions) is also sometimes called “within run precision” (ISO 3534-1, /6/). sr Repeatability standard deviation of a measurement (can be estimated from a series of duplicate analyses) R Reproducibility limit – performance measure for a test method or procedure when the test results are obtained under reproducibility conditions. Reproducibility conditions: Conditions where test results are obtained with the same method on identical test items in different laboratories with different operators using different equipment. Reproducibility (precision under reproducibility conditions) is also sometimes called “between lab precision” (ISO 3534-1, /6/). sR Reproducibility standard deviation of a measurement (can be estimated from validation studies with many participating laboratories or from other interlaboratory comparisons e.g. proficiency testing data) Note: R = 2.8 ⋅ s R Rw Within-laboratory reproducibility = intermediate measure between r and R, where operator and/or equipment and/or time and/or calibration can be varied, but in the same laboratory. An alternative name is intermediate precision sRw Reproducibility within-laboratory standard deviation (can be estimated from standard deviation of a control sample over a certain period of time, preferably one year) Page 1 of 41 CRM Certified Reference Material Certified value Assigned value given to a CRM, quantified through a certification process (traceable to SI-unit and with a known uncertainty) Nominal value Nominal value is the assigned value, e.g. in an interlaboratory comparison where it is the organiser’s best representation of the “true value” u(Cref) Uncertainty component from the certified or nominal value bias Difference between mean measured value from a large series of test results and an accepted reference value (a certified or nominal value). The measure of trueness is normally expressed in term of bias. Bias for a measurement, e.g. for a laboratory or for an analytical method.. Uncertainty component for bias. The u(bias), is always included in the measurement uncertainty calculations u(bias) RMSbias ∑ (bias i )2 n Interlaboratory General term for a collaborative study for either method comparison performance, laboratory performance (proficiency testing) or material certification. Page 2 of 41 2 Introduction 2.1 Scope and field of application This handbook is written for environmental testing laboratories in the Nordic countries, in order to give support to the implementation of the concept of measurement uncertainty for their routine measurements. The aim is to provide a practical, understandable and common way of measurement uncertainty calculations, mainly based on already existing quality control and validation data, according to the European accreditation guideline /12/, the Eurolab Technical Report No. 1 /3/ and the ISO/DTS 21748 Guide /8/. Nordtest has supported this project economically in order to promote and enhance harmonisation between laboratories on the Nordic market. Practical examples, taken directly from the everyday world of environmental laboratories, are presented and explained. However, the approach is very general and should be applicable to most testing laboratories in the chemical field. The handbook covers all steps in the analytical chain from the arrival of the sample in the laboratory until the data has been reported. It is important to notice that vital parts of the total measurement uncertainty are not included, e.g. sampling, sample transportation and possible gross errors during data storage/retrieval. The recommendations in this document are primarily for guidance. It is recognised that while the recommendations presented do form a valid approach to the evaluation of measurement uncertainty for many purposes, other suitable approaches may also be adopted – see references in Section 9. Especially the EURACHEM/CITAC-Guide /2/ is useful in cases where sufficient previous data is not available, and therefore the mathematical analytical approach according to GUM /1/ with all different steps is to be used. Basic knowledge in the use of quality control and statistics is required. In order to make it possible for the reader to follow the calculations, some raw data is given in appendices 2.2 Comment to customers Previously, laboratories usually reported uncertainty as the standard deviation calculated from data for an internal control sample. The measurement uncertainty also taking into account method and laboratory bias and using a coverage factor of 2, can give uncertainty values which may be a factor of 2 to 5 times higher than previously (Figure 1). However, this does not reflect a change in the performance of the laboratory, just a much better estimation of the real variation between laboratories. In Figure 1, the ammonium results from two laboratories are in good agreement – the difference is about 5 %. You can see this if you look to the right where measurement uncertainty is calculated correctly, but not if you look to the left, where the uncertainty is calculated directly from a control sample and presented as the standard deviation (± 1s). Page 3 of 41 250 Measurement uncertainty from... 240 Lab 1 230 Lab 2 Ammonium µg/l 220 210 200 190 180 Control sample Rw 170 Control sample + CRM or interlaboratory comparisons 160 150 Figure 1. Comparing ammonium results from two laboratories, Lab 1 = 199 µg/L and Lab6 0 2 = 188 µg/L. To the left the error bars are calculated from results on control samples (± 1s) and to the right the error bars are expanded measurement uncertainty. 2.3 About Measurement Uncertainty What is measurement uncertainty? ! ! ! The number after ± All measurements are affected by a certain error. The measurement uncertainty tells us what size the measurement error might be. Therefore, the measurement uncertainty is an important part of the reported result Definition: Measurement uncertainty is ”A parameter associated with the result of a measurement, that characterises the dispersion of the values that could reasonably be attributed to the measurand” /1, 5/ Who needs measurement uncertainties? ! ! The customer needs it together with the result to make a correct decision. The uncertainty of the result is important, e.g. when looking at allowable (legal) concentration limits The laboratory to know its own quality of measurement and to improve to the required quality Why should the laboratory give measurement uncertainty? ! ! As explained above, the customers need it to make correct decisions An estimation of the measurement uncertainty is required in ISO 17025 /9/ Page 4 of 41 How is measurement uncertainty obtained? ! ! ! The basis for the evaluation is a measurement and statistical approach, where the different uncertainty sources are estimated and combined into a single value “Basis for the estimation of measurement uncertainty is the existing knowledge (no special scientific research should be required from the laboratories). Existing experimental data should be used (quality control charts, validation, interlaboratory comparisons, CRM etc.)” /12/ Guidelines are given in GUM /1/, further developed in, e.g., EA guidelines /12/, the Eurachem/Citac guide /2/, in a Eurolab technical report /3/ and in ISO/DTS 21748 /8/ How is the result expressed with measurement uncertainty? ! ! ! Measurement uncertainty should normally be expressed as U, the combined expanded measurement uncertainty, using a coverage factor k = 2, providing a level of confidence of approximately 95 % It is often useful to state how the measurement uncertainty was obtained Example, where ± 7 is the measurement uncertainty: Ammonium (NH4-N) = 148 ± 7 µg/L. The measurement uncertainty, 7 µg/L (95 % confidence level, i.e. the coverage factor k=2) is estimated from control samples and from regular interlaboratory comparisons How should measurement uncertainty be used? ! ! It can be used as in Figure 1, to decide whether there is a difference between results from different laboratories, or results from the same laboratory at different occasions (time trends etc.) It is necessary when comparing results to allowable values, e.g. tolerance limits or allowable (legal) concentrations Page 5 of 41 3 Calculation of expanded uncertainty, U - overview A common way of presenting the different contributions to the total measurement uncertainty is to use a so-called fish-bone (or cause-and-effect) diagram. We propose a model (Figure 2), where either the reproducibility within-laboratory (Rw ) is combined with estimates of the method and laboratory bias, (error model in Appendix 3) or the reproducibility sR is used more or less directly, ISO Guide 21748/8/. The alternative way is to construct a detailed fish-bone diagram and calculate/estimate the individual uncertainty contributions. This approach may prove very useful when studying or quantifying individual uncertainty components. It has been shown, though, that in some cases this methodology underestimates the measurement uncertainty /3/, partly because it is hard to include all possible uncertainty contributions in such an approach. By using existing and experimentally determined quality control (QC) and method validation data, the probability of including all uncertainty contributions will be maximised. Measurement uncertainty model – fish-bone diagram covering the analytical process from sample arrival to report QC - Reproducibility within laboratory, Rw (section 4) Value Analytical Report Method and lab bias • Reference material • Interlab comparison • Validation (section 5) Decision maker Customer Reproducibility between laboratories sR (section 6) Figure 2. Measurement uncertainty model (fish-bone diagram), where the reproducibility within-laboratory is combined with estimates of the method and laboratory bias. Alternatively, according to ISO guide 21748 /8/, the combined uncertainty uc can be directly estimated from the reproducibility between laboratories (sR). This approach is treated in section 6 Page 6 of 41 3.1 Customer needs Before calculating or estimating the measurement uncertainty, it is recommended to find out what are the needs of the customers. After that, the main aim of the actual uncertainty calculations will be to find out if the laboratory can fulfil the customer demands with the analytical method in question. However, customers are not used to specifying demands, so in many cases the demands have to be set in dialogue with the customer. In cases where no demands have been established, a guiding principle could be that the calculated expanded uncertainty, U, should be approximately equal to, or less than, 2 times the reproducibility, sR. 3.2 Flow scheme for uncertainty calculations The flow scheme presented in this section forms the basis for the method outlined in this handbook. The flow scheme, involving 6 defined steps, should be followed in all cases. The example with NH4-N in water shows the way forward for calculating the measurement uncertainty using the flow scheme. Explanations of the steps and their components will follow in the succeeding chapters. For each step, there may be one or several options for finding the desired information. Background for the NH4-N example – automatic photometric method: The laboratory has participated in 6 interlaboratory comparisons recently. All results have been somewhat higher than the nominal value. The laboratory therefore concludes that there may be a small positive bias. On average, the bias has been +2.2 %. This bias is considered small by the laboratory and is not corrected for in their analytical results, but exists, and is thus another uncertainty component. For this method, the main sources of uncertainty are contamination and variation in sample handling, both causing random uncertainty components. These uncertainty sources will be included in the calculations below. Page 7 of 41 Step Action Specify Measurand Example – Ammonium NH4-N Ammonium is measured in water according to EN/ISO 11732 /11/. The customer demand on expanded uncertainty is ± 10 % Quantify Rw comp. A control sample B possible steps not covered by the control sample A: Control limits are set to ± 3.34 % (95 % confidence limit) 3 Quantify bias comp. From interlaboratory comparisons over the last 3 years the bias result were 2.4; 2.7; 1.9: 1.4; 1.8: and 2.9. The root mean square (RMS) of the bias is 2.25 %. The uncertainty of the nominal values is u(Cref) = 1.5 %. (see Appendix 4 for explanations) 4 Convert components to standard uncertainty u(x) Confidence intervals and similar distributions can be converted to standard uncertainty /1, 2, 14/. u(Rw) = 3.34/2 = 1.67 % 1 2 B: The control sample includes all analytical steps. u(bias) = RMSbias + u(Cref )2 2 = 2.252 + 1.5 = 2.71 % 5 Calculate combined standard uncertainty, uc Standard uncertainties can be summed by taking the square root of the sum of the squares uc = u(Rw ) 2 + (u(bias)) = 1.672 + 2.712 = 3.18 2 6 Calculate expanded uncertainty, U = 2 ⋅ uc The reason for calculating the expanded uncertainty is to reach a high enough confidence (app. 95 %) in that the “true value” lies within the interval given by the measurement result ± the uncertainty. U = 2 ⋅ 3.18 = 6.36 ≈ 6 %. The measurement uncertainty for NH4-N will thus be reported as ± 6 % at this concentration level. Page 8 of 41 3.3 Summary table for uncertainty calculations The results of the calculations done in the flow scheme will then be summarised in a summary table. Ammonium in water by EN/ISO 11732 Measurement uncertainty U (95 % confidence interval) is estimated to ± 6 %. The customer demand is ± 10 %. The calculations are based on control chart limits and interlaboratory comparisons. Value Relative u(x) Comments Reproducibility within-laboratory, Rw Control sample X Rw = 200 µg/L Other components Control limits is set to ± 3.34 % 1.67 % -- Method and laboratory bias Reference material bias -- Interlaboratory comparisons bias RMSbias= Recovery test bias 2.25 % u(Cref) = 1.5 % 2.71 % u(bias) = RMSbias + u(Cref )2 2 -- Reproducibility between laboratories Interlaboratory comparisons R -- Standard method R -- 8.8 % Data - see Section 6.2 Combined uncertainty, uc is calculated from the control sample limits and bias estimation from interlaboratory comparisons. The sR from interlaboratory comparisons can also be used (see 6.2) if a higher uncertainty estimation is acceptable. Measurand Ammonium Combined Uncertainty uc 1.67 2 + 2.67 2 = 3.18 % Expanded Uncertainty U 3.18 · 2 = 6.4 ≈ 6 % Page 9 of 41 4 Reproducibility within-laboratory - u(Rw) In this section the most common ways of estimating the reproducibility withinlaboratory component, u(Rw), for the measurement uncertainty calculation are explained: • Stable control samples covering the whole analytical process. Normally one sample at low concentration level and one at a high concentration level. • Control samples not covering the whole analytical process. Uncertainties estimated from control samples and from duplicate analyses of real samples with varying concentration levels. • Unstable control samples. It is of utmost importance that the estimation must cover all steps in the analytical chain and all types of matrices – worst-case scenario. The control sample data should be run in exactly the same way as the samples e.g. if the mean of duplicate samples is used for ordinary samples, then the mean of duplicate control samples should be used for the calculations. It is likewise important to cover long-term variations of some systematic uncertainty components within the laboratory, e.g. caused by different stock solutions, new batches of critical reagents, recalibrations of equipment, etc. In order to have a representative basis for the uncertainty calculations and to reflect any such variation the number of results should ideally be more than 50 and cover a time period of approximately one year, but the need differs from method to method. 4.1 Customer demands Some laboratories choose to use the customer demand when setting the limits in their control charts. The actual performance of the method is not interesting, as long as it meets the customer demands on expanded uncertainty. If, for example, the customer asks for data with an (expanded) measurement uncertainty of ± 10 %, then, from our experience, a good starting point is to set the control limits ± 5 %. The u(Rw) used in the calculations will then be 2.5 %.1 This is just a proposal and the measurement uncertainty calculations will show if these control limits are appropriate. 4.2 Control sample covering the whole analytical process When a stable control sample is covering the whole analytical process and has a matrix similar to the samples, the within-laboratory reproducibility at that concentration level can simply be estimated from the analyses of the control 1 Treating the control limits according to GUM /1/ as type B estimate with 95 % confidence limit Page 10 of 41 samples. If the analyses performed cover a wide range of concentration levels, also control samples of other concentration levels should be used. Example: For NH4-N two control sample levels (20 µg/L and 250 µg/L) were used during year 2002. The results for the manual analysis method are presented in the table below. Value Relative u(x) Comments Reproducibility within-laboratory, Rw Control sample 1 X = 20.01 µg/L sRw Control sample 2 X = 250.3 µg/L sRw Standard deviation 0.5 µg/L Standard deviation 3.7 µg/L -- Other components 2.5 % 1.5 % From measurements in 2002, n = 75 From measurements in 2002, n = 50 4.3 Control sample for different matrices and concentration levels When a synthetic control solution is used for quality control, and the matrix type of the control sample is not similar to the natural samples, we have to take into consideration uncertainties arising from different matrices. Example: To estimate the matrix based uncertainties, duplicate analysis of ammonium is performed, and the sr is estimated from the corresponding R%-chart (Range%-chart /13/), giving the repeatability of analysing natural samples having a normal matrix variation at different concentration levels. The data set consists of 73 duplicate analyses in the range of 2 µg/L – 16000 µg/L. Most of the results were below 200 µg/L. The data is divided into two parts: < 15 µg/L and > 15 µg/L The sr can be estimated from R%-charts constructed for both concentration ranges. The data is given in Appendix 5. The standard deviation is estimated from the range (see Appendix 8): s = range / 1.128 . Value Relative u(x) Comments Reproducibility within-laboratory, Rw Variation from duplicate analysis 2-15 µg/L: > 15 µg/L: sR 5.7 % 3.6 % n = 43 ( X = 6.50 µg/L) n = 30 ( X = 816 µg/L) Page 11 of 41 At low levels it is often better to use an absolute uncertainty rather than a relative, as relative numbers tend to become extreme at very low concentrations. In this example the absolute u(r) becomes 0.37 µg/L for the natural sample (mean concentration 7 µg/L) and 0.5 µg/L for the control sample in Section 4.2 (mean concentration 20 µg/L). As the estimate from duplicate analysis gives the repeatability component (sr) only, it should be combined with the control sample results from Section 4.2 to give a better estimate of sRw. This way, the repeatability component will be included two times, but it is normally small in comparison to the between-days variation. Value u(x) Comments Reproducibility within-laboratory, Rw Low level (2-15 µg/L) High level (> 15 µg/L) sRw sRw 0.5 µg/L from 0.6 µg/L control sample and 0.37 µg/L from duplicates 1.5% from 3.9 % control sample and 3.6% from duplicates Absolute u(x) = 0.5 2 + 0.37 2 Relative u(x) = 1.5 2 + 3.6 2 It can be noticed that the sample matrix has some effect on the variation of the results. The reason for this is not only the matrix, but also the concentration level, as almost all of the duplicate analyses were performed at a concentration level below 10 µg/L. The quantification limit of the measurement was 2 µg/L and the relative standard deviation usually becomes higher near that limit (cf. Figures 4 and 5 in Section 6.3). 4.4 Unstable control samples If the laboratory does not have access to stable control samples, the reproducibility can be estimated using analysis of natural duplicate samples. The results from the duplicate sample analysis can either be treated in an R-chart, where the difference between the first and second analysis is plotted directly, or as an R %-chart, where the absolute difference between the sample pair is calculated in % of the average value of the sample pair. The latter approach is particularly useful when the concentration varies a lot from time to time. In this example, duplicate samples for oxygen have been analysed on 50 occasions. The raw data is given in Appendix 6. The concentration variation is limited, so an R-chart approach is chosen. The difference between the first and the second analysis is calculated and plotted in a chart, see Figure 3. In this case, the second result is always subtracted from the first when constructing the R-chart, as it is important to look for systematic differences between the first and the second Page 12 of 41 sample. The standard deviation for the results can be estimated from the average range of the duplicate samples (see Appendix 8), and in this case becomes 0.024. The control limits at ±2s thus lies at ±0.048. The average value of the first determination is 7.53, and the sr thus equals 100·0.024/7.53 = 0.32 %. Oxygen in seawater, double samples Difference, mg/L 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 0 10 20 30 40 50 Analysis no. Figure 3. The difference between oxygen duplicate measurements plotted in an R-chart However, this only gives the within-day variation (repeatability, sr) for sampling and measurement, and there will also be a “long-term” uncertainty component from the variation in the calibration (here the thiosulphate used for titrating or the calibration of the oxygen probe, depending on method). For this particular analysis, the uncertainty component from the long-term variation in calibration is hard to measure, as no stable reference material or CRM is available for dissolved oxygen. One method would be to calibrate the same thiosulphate solution several times during a few days time, and use the variation between the results. Here we choose to estimate that component by a qualified guess, but laboratories are encouraged to also try the experimental approach. The total reproducibility within-laboratory for dissolved oxygen thus becomes: Page 13 of 41 Value Relative u(x) Comments Reproducibility within-laboratory, Rw Duplicate measurements of natural samples, difference used in rchart Estimated variation from differences in calibration over time sR s = 0.024 mg/L 0.32 % X = 7.53 mg/L s = 0.5 % 0.5 % Combined uncertainty for Rw Repeatability + reproducibility in calibration Page 14 of 41 0.32 2 + 0.5 2 = 0.59 % Measurements in 2000-2002, n= 50 Estimate, based on experience 5 Method and Laboratory bias – u(bias) In this chapter the most common ways of estimating the bias components will be outlined, namely the use of CRM, participation in interlaboratory comparisons (proficiency test) and recovery tests. Sources of bias should always be eliminated if possible. According to GUM /1/ a measurement result should always be corrected if the bias is significant and based on reliable data such as a CRM. However, even if the bias is zero, it has to be estimated and treated as an uncertainty component. In many cases the bias can vary depending on changes in matrix. This can be reflected when analysing several matrix CRMs, e.g. the bias could be both positive and negative. Examples are given and explained for the proposed calculations. For every estimation of the uncertainty from the method and laboratory bias, two components have to be estimated to obtain u(bias): 1) the bias (as % difference from the nominal or certified value) 2) the uncertainty of the nominal/certified value, u(Cref) or u(Crecovery) The uncertainty of the bias, u(bias) can be estimated by u (bias ) = RMSbias + u (Cref ) where RMSbias = 2 2 ∑ (bias ) 2 i n and if only one CRM is used also the sbias have to be included and u(bias) can the be estimated /14, 15/2 by 2 s u (bias ) = (bias ) + bias + u (Cref ) 2 n 2 5.1 Certified Reference Material Regular analysis of a CRM can be used to estimate the bias. The reference material should be analysed in at least 5 different analytical series (e.g. on 5 different days) before the values are used. In this example the certified value is 11.5 ± 0.5, with a 95 % confidence interval. Uncertainty component from the uncertainty of the certified value Step Step Convert the confidence The confidence interval is ± 0.5. Divide this by 1.96 interval to u(Cref) to convert it to standard uncertainty: 0.5/1.96 = 0.26 Convert to relative 100·(0.26/11.5) = 2.21% uncertainty u(Cref) Page 15 of 41 3 4 Quantify Method and laboratory bias Convert components to standard uncertainty u (x) bias = 100 ⋅ (11.9 - 11.5)/11.5 = 3.48 % sbias = 2.2 % (n = 12) u(Cref) = 2.21 % u (bias ) = (bias )2 + sbias 2 n + u (Cref ) 2 = 2 (3.48) + 2.2 + 2.212 = 4.2 % 12 2 If several CRM:s are used, we will get different values for the bias. The uncertainty of the bias estimation will be calculated in the following way (see also section 5.2). 3 Quantify Method and laboratory bias bias CRM1 is 3.48%, s=2,2 (n=12), u(Cref)=2.21 % bias CRM2 is -0.9% s=2,0 (n=7)), u(Cref)=1.8 % bias CRM3 is 2.4%, s= 2,8 (n=10), u(Cref)=1.8 % For the bias the RMSbias = 2.50 mean u(Cref)=1,9 % 4 Convert components to standard uncertainty u (x) u (bias ) = RMS bias + u (Cref ) 2 Page 16 of 41 2 2.50 2 + 1.92 = 3.1% 5.2 Interlaboratory comparisons In this case the results from interlaboratory comparisons are used in the same way as a reference material, i.e. to estimate the bias. In order to have a reasonably clear picture of the bias from interlaboratory comparison results, a laboratory should participate at least 6 times within a reasonable time interval. Biases can be both positive and negative. Even if the results appear to give positive biases on certain occasions and negative on others, all bias values can be used to estimate the uncertainty component, RMSbias. The way forward is very similar to that for reference materials. However, the estimation of the bias from interlaboratory comparisons has more uncertainty to it, and thus usually becomes a bit higher than if CRMs are used. This is partly due to the fact that the certified value of a CRM normally is better defined than a nominal or assigned value in an interlaboratory comparison exercise. In some cases the calculated uncertainty u(Cref) from an interlaboratory comparison becomes too high and is not valid for estimating the u(bias). Uncertainty component from the uncertainty of the nominal value Step Example Find the between laboratory The sR has been on average 9% in the 6 exercises. standard deviations, sR, for the exercises. Calculate u(Cref) Mean number of participants = 12. u (Cref ) = sR 9 = = 2.6 % n 12 The bias has been 2 %, 7 %, -2 %, 3 %, 6 % and 5%, in the 6 interlaboratory comparisons where the laboratory has participated. 3 Quantify Method and laboratory bias 4 Convert components to standard uncertainty u (x) = 4.6 %, u(Cref)= 2.6 % RMSbias u(bias) = RMSbias + u(Cref ) 2 = 2 = 4.62 + 2.62 = 5.3% Page 17 of 41 5.3 Recovery Recovery tests, for example the recovery of a standard addition to a sample in the validation process, can be used to estimate the systematic error. In this way, validation data can provide a valuable input to the estimation of the uncertainty. In an experiment the recoveries for an added spike were 95 %, 98 %, 97 %, 96 %, 99 % and 96 % for 6 different sample matrices. The average is 96.8 %. The spike of 0.5 mL was added with a micropipette. Uncertainty component from the definition of 100% recovery, u(Crecovery) Step Example Uncertainty of the 100% u(conc) - Certificate ± 1.2 % (95 % conf. limit) gives = recovery. Main components 0.6 % are concentration, u(conc) u(vol) - This value can normally be found in the of standard and volume manufacturer’s specifications, or better use the limits added u(vol) specified in your laboratory. Max bias 1 % (rectangular interval) and repeatability max 0.5 % 2 1 u (vol ) = + 0.5 2 = 0.76 % 3 Calculate u(Crecovery) u (conc) 2 + u (vol ) 2 = 0.6 2 + 0.76 2 = 1.0 % = 3.44 % u(Crecovery)=1.0 % 3 Quantify Method and laboratory bias RMSbias 4 Convert components to standard uncertainty u (x) u (bias ) = RMS bias + u (Cref ) 2 = Page 18 of 41 2 = 3.44 2 + 1.0 2 = 3.6 % 6 Reproducibility between laboratories, sR If the demand on uncertainty is low, it can be possible to directly use the sR from interlaboratory comparisons as an approximation of uc /8/. In such case the expanded uncertainty becomes U = 2 ⋅ s R . This may be an overestimate depending on the quality of the laboratory – worst-case scenario. It may also be an underestimate due to sample inhomogenity or matrix variations. 6.1 Data given in standard method In order to use a figure taken directly from the standard method, the laboratory must prove that they are able to perform in accordance with the standard method /8/, i.e. demonstrating control of bias and verification of the repeatability, sr. Reproducibility data can either be given as a standard deviation sR or as reproducibility limit R and then sR = R/2.8 The example below is taken from ISO/DIS 15586 Water Quality — Determination of trace elements by atomic absorption spectrometry with graphite furnace. The matrix is wastewater. Combined uncertainty in wastewater, uc, is taken from the sR from interlaboratory comparison exercises quoted in the ISO method. Table 1 ISO/DIS 15586 - Results from the interlaboratory comparison – Cd in water with graphite furnace AAS. The wastewater was digested by the participants. n Outliers Nominal value Mean Recovery, Cd µg/L µg/L % sr sR % % Synthetic Lower 33 1 0.3 0.303 101 3.5 17.0 Synthetic Higher 34 2 2.7 2.81 104 1.9 10.7 Fresh water Lower 31 2 0.572 2.9 14.9 Fresh water Higher 31 3 3.07 2.1 10.4 Waste water 2 1.00 3.1 27.5 Measurand Cd 27 Combined Uncertainty uc uc = 27.5 % Expanded Uncertainty U 2·uc = 55 % ≈ 50 % 6.2 Data from interlaboratory comparisons Interlaboratory comparisons are valuable tools in uncertainty evaluation. The reproducibility between laboratories is normally given directly in reports from the exercises as sR. These data may well be used by a laboratory (having performed satisfactorily in the comparisons) as the standard uncertainty of the analysed parameter, provided that the comparison covers all relevant uncertainty components and steps (see /9/, Page 19 of 41 section 5.4.6.3). For example, a standard deviation in an interlaboratory comparison, sR, can be directly used as a combined standard uncertainty, uc. Table 2 Summary results (mean values) from 10 interlaboratory comparisons that Lab A has participated in. The reproducibility standard deviation is given in absolute units, sR and in relative units sR %. Variable pH Conductivity, mS/m Alkalinity, mmol/L Turbidity, FNU NH4-N, µg/L NO3-N, µg/L Nominal value 7.64 12.5 Lab A % deviation -0.037 -2.8 0.673 1.4 146 432 2.3 -9.1 2.2 -1.6 sR sR % 0.101 0.40 3.2 No. of labs 90 86 5 6 0.026 0.1 12.0 16.3 60 44 34 39 3 3 5 6 (abs) 3.9 14.2 8.8 3.7 Excluded In Table 2 we find that for conductivity, for instance, the mean value for the results from 10 interlaboratory comparisons is 12.5 mS/m. The reproducibility relative standard deviation is 0.4 (3.2 %), which is an average (or pooled) standard deviation between the laboratories in the different interlaboratory comparisons and this value can be taken as an estimate of combined uncertainty i.e. uc (conductivity) = sR = 0.4 mS/m, thus U = 2·0.4 = 0.8 mS/m If we take the ammonium results, we have a mean nominal value of 146 µg/L, and we find that the reproducibility, sR, is 8.8 %. Thus U = 2·8.8 = 17.6 = 18 % at this concentration level. Comment: In Section 3 the expanded uncertainty for ammonium is 5 % using an automated method in one highly qualified laboratory. mS/m Page 20 of 41 7 Examples In this chapter, practical examples on how measurement uncertainty can be calculated using the method of this handbook are given. 7.1 Ammonium in water Ammonium in water has already been treated in section 3.2 and section 6.2 . The results are summarised in Table 3. Table 3 Measurement uncertainty of ammonium in water – comparison of different calculations Uncertainty calculations based on Control sample + proficiency testing Interlaboratory comparisons Relative expanded uncertainty, U ±6% ± 18 % Comment Uncertainty for one good laboratory- level 200 µg/L. Uncertainty in general among laboratories – level 150 µg/L 7.2 BOD in wastewater Biological Oxygen Demand, BOD, is a standard parameter in the monitoring of wastewater. This example shows how data from ordinary internal quality control can be used together with CRM results or data from interlaboratory comparison exercises to calculate the within-lab reproducibility and bias components of the measurement uncertainty. The results are summarised in Table 4 Table 4 Measurement uncertainty of BOD in water - comparison of different calculations Uncertainty calculations based on Control sample + CRM Control sample + interlaboratory comparisons Interlaboratory comparisons Relative expanded uncertainty, U ± 10 % ± 10 % ± 16 % Comment n = 3, unreliable estimate Uncertainty in general among laboratories For BOD at high concentrations, using the dilution analytical method, the major error sources are the actual oxygen measurement and variation in the quality of the seeding solution. These errors will be included in the calculations. The raw data from the internal quality control, using a CRM, used for the calculations is shown in Appendix 7. Page 21 of 41 The laboratory has only participated in three interlaboratory comparison exercises the last 2 years (Table 5). At least six would be needed, so here we estimate the bias two different ways – with CRM and with interlaboratory comparisons. Table 5 BOD - results from interlaboratory comparisons Exercise 1 2 3 X RMSbias 3 Nominal value mg/L 154 219 176 Laboratory result mg/L 161 210 180 Bias sR Number of labs % + 4.5 - 4.1 +2.3 +0.9 3.76 % 7.2 6.6 9.8 7.873 - 23 25 19 22.3 - If sR or the number of participants vary substantially from exercise to exercise, then a pooled standard deviation will be more correct to use. In this case, where the variation in sR is limited, we simply calculate the mean sR (the corresponding pooled standard deviation becomes 7.82, an insignificant difference). Page 22 of 41 Example A: BOD with Internal quality control + a CRM Step 1 2 Action Specify Measurand Example: BOD in wastewater BOD in wastewater, measured with EN1899-1 (method with dilution, seeding and ATU). The demand on uncertainty is ± 20 %. Quantify u(Rw) A: The control sample, which is a CRM, gives an s = 2.6 % at a level of 206 mg/L O2. s = 2.6 % is also when setting the control chart limits. A control sample B possible steps not covered by the control sample 3 Quantify Method and laboratory bias B: The analysis of the control sample includes all analytical steps after sampling The CRM is certified to 206 ±5 mg/L O2. The average result of the control chart is 214.8. Thus, there is a bias of 8.8 mg/L = 4.3 %. The sbias is 2.6 % (n=19) The u(Cref) is 5 mg/L / 1.96 = 1.2 % 4 Convert components to standard uncertainty u(x) u(Rw) = 2.6 % 2 u(bias) = bias2 + sbias + u(Cref )2 n 2 2.6 = 4.32 + + 1.22 = 4.5 % 19 5 6 Calculate combined standard uncertainty, uc Calculate expanded uncertainty, U = 2 ⋅ u c uc = 2.6 2 + 4.5 2 = 5.2 % U = 2 ⋅ 5.2 = 10.4 ≈ 10 % Page 23 of 41 Example B: BOD with Internal quality control + interlaboratory comparison results Step 1 2 Action Specify Measurand Example: BOD in wastewater BOD in wastewater, measured with EN1899-1 (method with dilution, seeding and ATU). The demand on uncertainty is ± 20 %. Quantify u(Rw) A: The control sample, which is a CRM, gives an s of 2.6 % at a level of 206 mg/L O2. s = 2.6 % is also used as s when setting the control chart limits. A control sample B possible steps not covered by the control sample 3 4 B: The analysis of the control sample includes all analytical steps after sampling Quantify Method and laboratory bias Data from Table 5 RMSbias = 3.76 Convert components to standard uncertainty u(x) u(Rw) = 2.6 % u (Cref ) = sR 7.9 = = 1.67 n 22.3 u(bias) = RMSbias + u(Cref)2 = 2 3.762 +1.672 = 4.11% 5 6 Calculate combined standard uncertainty, uc Calculate expanded uncertainty, U = 2 ⋅ u c Page 24 of 41 uc = 2.6 2 + 4.112 = 4.86 % U = 2 ⋅ 4.86 = 9.7 ≈ 10 % 7.3 PCB in sediment In this example, the u(Rw) is estimated from a quality control sample and the u(bias) is estimated from two different sources: in the first example the use of a CRM and in the second example participation in interlaboratory comparisons. In the summary table both ways of calculating the u(bias) will be compared. For this analysis, the sample-work up is a major error source (both for random and systematic errors), and it is thus crucial that this step is included in the calculations. The number of interlaboratory comparisons is too few to get a good estimate. Example C: PCB with Internal quality control + a CRM Step 1 2 Action Specify Measurand Example: PCB in sediment Sum of 7 PCB:s in sediment by extraction and GC-MS(SIM). Demand on expanded uncertainty is ± 20 %. Quantify u(Rw) A: The control sample, which is a CRM, gives an sRw = 8 % at a level of 150 µg/kg dry matter. sRw = 8 % is also used when setting the control chart limits. B: The analysis of the control sample includes all steps except for drying the sample to determine the dry weight. The uncertainty contribution from that step is considered small and is not accounted for. A control sample B possible steps not covered by the control sample 3 Quantify method and laboratory bias The CRM is certified to 152 ± 14 µg/kg. The average result of the control chart is 144. Thus, there is a bias = 5.3 %. The sbias = 8 % (n=22) u(Cref) 14 µg/kg/1.96, which is 4.7 % relative. 4 Convert components to standard uncertainty u(x) u(Rw) = 8 % 2 u(bias) = bias2 + sbias + u(Cref )2 n 2 8 = 5.32 + + 4.72 = 7.29 22 5 Calculate combined standard uncertainty, uc uc = 8 2 + 7.29 2 = 10.8 % 6 Calculate expanded uncertainty, U = 2 ⋅ u c U = 2 ⋅ 10.8 = 21.6 ≈ 22 % Page 25 of 41 Example D: PCB with Internal quality control + interlaboratory comparison Step 1 2 Action Specify Measurand Example: PCB in sediment Sum of 7 PCB:s in sediment by extraction and GC-MS(SIM). Demand on expanded uncertainty is 20 %. Quantify u(Rw) A: The control sample, which is a stable inhouse material, gives sRw = 8 % at a level of 150 µg/kg dry matter. sRw = 8 % is also used as s when setting the control chart limits. B: The analysis of the control sample includes all steps except for drying the sample to determine the dry weight. The uncertainty contribution from that step is considered small and is not accounted for. A control sample B possible steps not covered by the control sample 3 Quantify Method and laboratory bias Participation in 3 interlaboratory comparisons with concentration levels similar to the internal quality control. The bias in the 3 exercises has been –2 %, -12 % and –5 %. RMSbias = 7.6 The sR in the three exercises has been 12 %, 10 % and 11 %, on average sR = 11 % (n=14) u (Cref ) = 4 Convert components to standard uncertainty u(x) 11 14 = 2.9 % The u(Rw) is 8 % u(bias) = RMSbias + u(Cref)2 = 2 7.62 + 2.92 = 8.1% 5 Calculate combined standard uncertainty, uc 6 Calculate expanded uncertainty, U = 2 ⋅ u c Page 26 of 41 uc = 8 2 + 8.12 = 11.4 U = 2 ⋅11.4 = 22.8 ≈ 23 % Summary table for PCB measurement uncertainty calculations PCB in sediment by extraction and GC-MS (SIM) Measurement uncertainty U (95 % confidence interval) is estimated to ± 20 % (relative) for 7 PCB:s in sediments at a level of 150 µg/kg dry weight. The customer demand is ± 20 %. The calculations are based on internal quality control using a stable sample, CRM and the participation in a limited amount of interlaboratory comparison exercises. Value u(x) Comments Reproducibility within-laboratory, Rw Control sample X = 160 µg/kg dry weight Other components u(Rw) 12.8 µg/kg dry weight 8% too small to be considered Method and laboratory bias Bias: 5.3 % Reference material u(bias) = 5.88 u(bias)= sbias = 8 ; n = 22 Interlaboratory comparison n=3 u(Cref) = 4,7 % RMSbias = 7.6 u(Cref) = 2.9 % bias 2 + u(bias) = 8.1 s bias n 2 + u(Cref ) 2 u(bias)= RMS bias + u (Cref ) 2 2 Combined uncertainty, uc, is calculated from internal quality control and the maximum bias - interlaboratory comparisons. Measurand PCB Combined Uncertainty uc uc = 8 2 + 8.12 = 11.4 Expanded Uncertainty U U = 2 ⋅ u c = 2 ⋅ 11.4 = 22.8 ≈ 23 % Conclusion: In this case the calculation of the u(bias) gives similar results regardless of whether CRM or interlaboratory comparison results are used. Sometimes interlaboratory comparisons will give considerably higher values, and it might in such cases be more correct to use the CRM results. Page 27 of 41 7.4 Concentration ranges The measurement uncertainty will normally vary with concentration, both in absolute and relative terms. If the concentration range of the reported data is large, it is thus often necessary to take this into account. For lead (Pb) in water, a recovery experiment was carried out a number of times to investigate within-lab reproducibility over the measurable range – the major component of the measurement uncertainty at low levels. The following results were obtained: Table 6 Within-lab reproducibility and recovery for Pb determined with ICP-MS at different concentration levels. Addition, µg/L 0.01 0.1 0.4 1 10 10 100 Pb, % recovery 109.7 125.2 91.8 98.4 98 100.5 105.5 s, % 53.8 12.1 5 3.0 1.7 1.3 1.4 60 50 s, % 40 30 20 10 0 0.01 0.1 1 10 100 Concentration (log scale) Figure 4 Within-lab reproducibility for Pb over the concentration range It is clear from the results that the measurement uncertainty, here represented by s, is strongly concentration dependent. Two approaches are recommended for using these data: (1) To divide the measurable range into several parts, and use a fixed relative measurement uncertainty or absolute uncertainty – see Table 7. Page 28 of 41 Table 7 Within-lab reproducibility for Pb divided into three concentration ranges Within-lab reproducibility Pb Range (µg/L) s(rel) s(rel) or (abs) < 0.1 50 % 0.01 (µg/L) 0.1 - 10 10 % 10 % > 10 2% 2% In the second column s is relative and given in %. In the third column s is also relative but an absolute value is given in the lower range close to the detection limit. (2) To use an equation that describes how the measurement uncertainty varies with concentration Plotting s % against 1/concentration gives a straight line, and a relatively simple equation. (see Figure 5). 15 s 10 % 5 y = 1.06x + 1,77 2 R = 0.9798 0 0 2 4 6 8 10 12 1/concentration Figure 5: The relationship between within-lab reproducibility and the inverted concentration for Pb in the range 0.1 – 100 µg/L. The straight-line equation above tells us that the within-lab reproducibility equals 1.06 multiplied with 1/concentration plus 1.77. For example, at a concentration of 2 µg/L the within-lab reproducibility becomes 1.06·1/2 + 1.77 = 2.3 %. When reporting to customers, the laboratory can choose between quoting the formula or calculating the measurement uncertainty for each value, using the formula. For further reading, see for example /2/. Page 29 of 41 8 Reporting uncertainty This is an example on what a data report could look like, when measurement uncertainty has been calculated and is reported together with the data. The company and accreditation body logotypes are omitted, and the report does not contain all information normally required for an accredited laboratory. It is recommended to use either relative or absolute values for the benefit of the customer. Analytical Report Sample identification: P1 – P4 Samples received: 14 December 2002 Analysis period: 14 –16 December 2002 Results NH4-N (µg/L): Sample P1 P2 P3 P4 Result 103 122 12 14 U ±6% ±6% ±10% ±10% Method 23B 23B 23B 23B TOC (mg/L) Sample P1 P2 P3 P4 Result 40 35 10 9 U ±4.0 ±3.5 ±1.0 ±0.9 Method 12-3 12-3 12-3 12-3 Signed: Dr Analyst Page 30 of 41 The laboratory should also prepare a note explaining how the measurement uncertainty has been calculated for the different parameters. Normally, such an explanatory note should be communicated to regular customers and other customers who ask for information. An example is given below: Note on measurement uncertainty from Dr Analyst’s laboratory Measurement uncertainty: U = expanded Measurement Uncertainty, estimated from control sample results, interlaboratory comparison and the analyses of CRMs, using a coverage factor of 2 to reach approximately 95% confidence level. NH4-N: U is estimated to 6% above 100 µg/L and 10% below 100 µg/L. TOC: U is estimated to 10% over the whole concentration range. References: • Guide To The Expression Of Uncertainty In Measurement (GUM) • Quantifying Uncertainty in Analytical Measurement. EURACHEM/CITAC Guide • Handbook for calculation of measurement uncertainty in environmental laboratories Page 31 of 41 9 References 1. Guide To The Expression Of Uncertainty In Measurement (GUM). BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML. International Organization of Standardization, Geneva Switzerland, 1st Edition 1993, Corrected and reprinted 1995. 2. Quantifying Uncertainty in Analytical Measurement. EURACHEM/CITAC Guide, 2nd Edition, 2000 3. Measurement Uncertainty in Testing, Eurolab Technical Report No. 1/2002 4. Interlaboratory comparison test data, personal communication, H. Hovind, NIVA, Norway. 5. International Vocabulary of Basic and General Terms in Metrology (VIM). ISO, 1993 6. ISO/IEC 3534-1-2, Statistics – Vocabulary and symbols Parts 1-2 7. ISO 5725-1-6:1994, Accuracy (trueness and precision) of measurement methods and results 8. ISO/DTS 21748:2003, Guide to the use of repeatability, reproducibility and trueness estimates in measurement uncertainty estimation 9. EN ISO/IEC 17025:2000, General Requirements for the Competence of Calibration and Testing Laboratories 10. ISO/TR 13530:1997, Water quality – Guide to analytical quality control for water analysis 11. EN ISO 11732:1997, Water quality -- Determination of ammonium nitrogen by flow analysis (CFA and FIA) and spectrometric detection 12. EA-4/xx: Proposed “EA guideline on The Expression of uncertainty in quantitative testing”. Project to be finalised in 2003. (www.europeanaccreditation.org/) 13. ISO8258, First edition, 1991-12-15, Shewhart Control Charts 14. V. J. Barwick, Ellison L.R., Analyst, 1999, 124, 981-990 15. E. Hund, D.L. Massart and J. Smeyers-Verbeke, Operational definitions of uncertainty. TrAC, 20 (8), 2001 Page 32 of 41 10 Appendices Appendix 1: Empty flow scheme for calculations Before starting: Always identify the main error sources, to make sure that they are included in the calculations. Step 1 2 Action Specify Measurand Measurand: (measurand) in (matrix) by (method) The customer demand on expanded uncertainty is ± _ %. Quantify u(Rw) A control sample B possible steps not covered by the control sample A: 3 Quantify method and laboratory bias 4 Convert components to standard uncertainty u(x) 5 Calculate combined standard uncertainty, uc 6 Calculate expanded uncertainty, U = 2 ⋅ u c B: Page 33 of 41 Appendix 2: Empty summary table (measurand) in (matrix) by (method) Measurement uncertainty U (95 % confidence interval) is estimated to ± _ % (relative) for (measurand) in (matrix) at a level of _ (unit). The customer demand is ± _ %. The calculations are based on (control samples/control limits/CRM/interlaboratory comparison/other). Value Relative u(x) Comments Reproducibility within-laboratory, Rw Control sample X sRw = (conc) (unit) Other components Method and laboratory bias Reference material bias Interlaboratory comparison bias Recovery test bias Reproducibility between laboratories Interlaboratory comparison sR Standard method sR Combined uncertainty, uc, is calculated from __ and bias from __. Measurand Combined Uncertainty uc Expanded Uncertainty U 2 ⋅ uc = Page 34 of 41 Appendix 3: Error model used in this handbook This model is a simplification of the model presented in the ISO guide /8/: y = m + (δ + B) + e y measurement result of a sample m expected value for y δ method bias B laboratory bias – the uncertainty for these are combined to u(bias) e random error at within-laboratory reproducibility conditions, Rw Uncertainty estimation in section 3 to 5 u ( y ) 2 = s Rw + u (bias ) 2 2 s Rw 2 The estimated variance of e under within-laboratory reproducibility conditions – intermediate precision. In the ISO guide the repeatability, sr is used as an estimate of e. u (bias ) 2 The estimated variance of method bias and laboratory bias. Uncertainty estimation in section 6 The combined uncertainty u(y) or uc can also be estimated by from reproducibility data. u ( y ) 2 = s L + s r = s R - equation A6 ref. /8/ 2 2 2 where sR2 is the estimated variance under reproducibility conditions and where sL2 is either the estimated variance of B if one method is used by all laboratories or an estimated variance of B and δ if several different methods have been used in the collaborative study and sr2 is the estimated variance of e. Comment For samples that are more inhomogeneous and have big variations in matrix the estimation of the measurement uncertainty of the method can become too low. However we recommend the use of repeatability limit for duplicate analyses r = 2.8 ⋅ s r in order to control sample inhomogenity. Page 35 of 41 Appendix 4: Uncertainty of bias for NH4-N in section 3.2 Results for a laboratory from interlaboratory comparisons of NH4-N in water. Exercise Nominal value xref mg/L 1999 1 2 2000 1 2 2001 1 2 Laboratory result xi mg/L 81 73 264 210 110 140 83 75 269 213 112 144 X RMS RMS of the bias = u(Cref) = sR n = ∑ bias n 2 i = Bias sR % % 2.4 2.7 1.9 1.4 1.8 2.9 + 2.18 2.25 10 7 8 10 7 11 8.8 - Number of labs 31 36 32 35 36 34 34 - 2.4 2 + 2.7 2 + ...2.9 2 = 2.25 % (rel) n 8.8 = 1.5 % (rel) 34 A t-test shows that the bias (+2.18 %) is not significant (t = 0.6). However, in order not to complicate the calculations when the bias is small, t-test are normally not performed. The mean value of sR is used. If differences in number of laboratories and sR are very big pooled standard deviations should be used. In this case the pooled standard deviation is 8.9 % for sR which is the same as the mean value of 8.8 %. Page 36 of 41 Appendix 5: Raw data for NH4-N in section 4.3 The estimation of the standard deviation from the range is explained in Appendix 8 concentration < 15 µg/L Sam ple 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 X1 7.46 9.01 3.6 6.48 14.49 10.84 4.61 2.6 2.8 5.84 2.12 2.3 2.52 3.71 7.43 8.83 9.12 8.24 2.62 3.33 2.69 12.09 4.24 10.49 3.68 9.37 2.22 6.1 2.96 14.02 4.24 5.1 2.78 8.52 12.82 3.17 11.28 14.31 4.01 3.27 9.98 12.56 3.35 X2 7.25 9.17 3.1 6.48 14.12 9.89 5 2.42 2.62 6.19 2.5 2.11 2.89 3.71 7.43 8.51 8.79 7.9 2.78 3.33 2.69 12.09 4.24 10.64 3.52 9.37 2.06 6.1 2.86 13.7 3.62 4.61 2.62 6.81 14.05 2.4 11.43 13.82 4.48 3.58 10.29 13.66 2.88 M ean: 7.355 9.090 3.350 6.480 14.305 10.365 4.805 2.510 2.710 6.015 2.310 2.205 2.705 3.710 7.430 8.670 8.955 8.070 2.700 3.330 2.690 12.090 4.240 10.565 3.600 9.370 2.140 6.100 2.910 13.860 3.930 4.855 2.700 7.665 13.435 2.785 11.355 14.065 4.245 3.425 10.135 13.110 3.115 6.499 0.210 -0.160 0.500 0.000 0.370 0.950 -0.390 0.180 0.180 -0.350 -0.380 0.190 -0.370 0.000 0.000 0.320 0.330 0.340 -0.160 0.000 0.000 0.000 0.000 -0.150 0.160 0.000 0.160 0.000 0.100 0.320 0.620 0.490 0.160 1.710 -1.230 0.770 -0.150 0.490 -0.470 -0.310 -0.310 -1.100 0.470 s(r) % = range(mean)/1.128 = 2.855 1.760 14.925 0.000 2.587 9.165 8.117 7.171 6.642 5.819 16.450 8.617 13.678 0.000 0.000 3.691 3.685 4.213 5.926 0.000 0.000 0.000 0.000 1.420 4.444 0.000 7.477 0.000 3.436 2.309 15.776 10.093 5.926 22.309 9.155 27.648 1.321 3.484 11.072 9.051 3.059 8.391 15.088 6.4363 5.71 = m ean range (% ) % Page 37 of 41 concentration > 15 µg/L X1 Sample 1 37.62 2 16.18 3 28.82 4 4490 5 135.7 6 62.56 7 158.9 8 16540 9 31.26 10 58.49 11 740.5 12 130.3 13 29.35 14 1372 15 36.55 16 22.57 17 34.75 18 92.93 19 40.6 20 80.36 21 15.76 22 78.22 23 48.89 24 17.65 25 36.56 26 51.89 27 197.5 28 70.32 29 29.99 30 31.9 X2 36.85 37.235 16.56 16.370 28.65 28.735 4413 4451.500 124.7 130.200 62.25 62.405 159.2 159.050 16080 16310.000 30.12 30.690 60.11 59.300 796.2 768.350 126.9 128.600 29.19 29.270 1388 1380.000 44.74 40.645 23.37 22.970 33.15 33.950 94.01 93.470 42.23 41.415 86.36 83.360 18.54 17.150 73.76 75.990 50.91 49.900 16.72 17.185 35.3 35.930 52.2 52.045 206.5 202.000 69.22 69.770 30.62 30.305 32.36 32.130 Mean: 816.331 0.770 -0.380 0.170 77.000 11.000 0.310 -0.300 460.000 1.140 -1.620 -55.700 3.400 0.160 -16.000 -8.190 -0.800 1.600 -1.080 -1.630 -6.000 -2.780 4.460 -2.020 0.930 1.260 -0.310 -9.000 1.100 -0.630 -0.460 s(r) % = range(mean)/1.128 = Page 38 of 41 2.068 2.321 0.592 1.730 8.449 0.497 0.189 2.820 3.715 2.732 7.249 2.644 0.547 1.159 20.150 3.483 4.713 1.155 3.936 7.198 16.210 5.869 4.048 5.412 3.507 0.596 4.455 1.577 2.079 1.432 4.0843 3.62 = mean range (%) % Appendix 6: Raw data for oxygen in Section 4.4 Data plotted in Figure 3. “Range” equals the absolute value of the difference between Result 1 and Result 2. R e s . 1 m g /L 8 .9 0 8 .9 9 8 .9 0 9 .1 1 8 .6 8 8 .6 0 8 .8 1 8 .0 2 7 .0 5 6 .9 8 7 .1 3 6 .7 9 6 .5 5 4 .6 8 5 .2 8 7 .4 2 7 .6 2 5 .8 8 6 .0 3 6 .3 3 5 .9 0 6 .2 4 6 .0 2 9 .1 3 9 .1 0 8 .5 0 8 .7 3 8 .0 9 7 .5 6 6 .3 0 6 .4 3 7 .2 5 7 .2 8 8 .0 0 8 .3 8 9 .2 3 9 .0 9 9 .3 7 9 .3 8 9 .3 2 8 .4 7 8 .2 7 8 .3 7 8 .0 9 8 .0 5 7 .3 8 7 .4 9 4 .5 2 4 .4 5 4 .2 9 m e a n R e s . 2 m g /L 8 .9 1 9 .0 1 8 .9 0 9 .1 2 8 .6 4 8 .5 1 8 .8 1 8 .0 0 7 .0 8 7 .0 1 7 .1 6 6 .7 8 6 .5 3 4 .6 8 5 .3 3 7 .4 0 7 .6 3 5 .8 8 6 .0 6 6 .3 3 5 .9 0 6 .2 7 6 .0 2 9 .1 1 9 .1 4 8 .4 4 8 .7 1 8 .0 9 7 .5 8 6 .3 2 6 .4 4 7 .3 4 7 .3 1 8 .0 3 8 .2 9 9 .2 9 9 .0 8 9 .3 6 9 .3 7 9 .2 5 8 .4 9 8 .2 8 8 .3 1 8 .1 5 8 .0 3 7 .4 0 7 .4 9 4 .4 9 4 .4 4 4 .2 7 m e a n ra n g e : r a n g e /1 .1 2 8 : R a n g e m g /L 0 .0 1 0 .0 2 0 .0 0 0 .0 1 0 .0 4 0 .0 9 0 .0 0 0 .0 2 0 .0 3 0 .0 3 0 .0 3 0 .0 1 0 .0 2 0 .0 0 0 .0 5 0 .0 2 0 .0 1 0 .0 0 0 .0 3 0 .0 0 0 .0 0 0 .0 3 0 .0 0 0 .0 2 0 .0 4 0 .0 6 0 .0 2 0 .0 0 0 .0 2 0 .0 2 0 .0 1 0 .0 9 0 .0 3 0 .0 3 0 .0 9 0 .0 6 0 .0 1 0 .0 1 0 .0 1 0 .0 7 0 .0 2 0 .0 1 0 .0 6 0 .0 6 0 .0 2 0 .0 2 0 .0 0 0 .0 3 0 .0 1 0 .0 2 0 .0 2 6 0 .0 2 4 Page 39 of 41 Appendix 7: Raw data for BOD in example A and B Results in mg/L O2 consumption. The certified value of the CRM is 206 ± 5 mg/L. As the average of two results is always reported for ordinary samples, the s is also calculated from the average of each sample pair in the internal quality control. Date 12-09-00 01-03-01 13-03-01 02-04-01 14-08-01 05-09-01 19-09-01 16-10-01 07-11-01 28-11-01 11-12-01 13-12-01 15-01-02 22-01-02 30-01-02 11-02-02 06-03-02 18-09-02 02-10-02 Page 40 of 41 Res. 1 218.90 206.46 221.18 215.00 194.96 218.65 223.86 215.58 196.26 210.89 228.40 206.73 207.00 224.49 201.09 218.83 216.69 206.36 215.21 Res. 2 Average 214.77 216.84 220.83 213.65 210.18 215.68 206.50 210.75 218.03 206.50 216.55 217.60 212.19 218.03 213.01 214.30 214.93 205.60 206.89 208.89 222.73 225.57 229.03 217.88 208.47 207.74 213.66 219.08 214.07 207.58 223.13 220.98 218.22 217.46 227.96 217.16 226.18 220.70 Average: 214.84 s: 5.58 s%: 2.60 Appendix 8: Estimation of standard deviation from range Number of samples n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 Factor , d2 1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078 Estimation of standard deviation from range (max-min), /1/ and /13, page 11/. The standard deviation, s can be estimated from s= range d2 For comparison Rectangular interval 95 % conf. limit. 3.464 3.92 (Example, see Appendix 5 and 6) Page 41 of 41 TECHNICAL REPORTS FROM EXPERT GROUP QUALITY AND METROLOGY 351 Guðmundsson, H., Calibration and uncertainty of XRF measurements in the SEM. Espoo 1997. Nordtest, NT Techn Report 351. 42p. NT Project No. 1280-96. 352 Rydler, K.-E. & Nilsson, H., Transparency of national evaluation methods of accreditation in the field of electrical measurements. Espoo 1997. Nordtest, NT Techn Report 352. 61p. NT Project No. 1255-95. 353 Tambo, M. & Søgaard, T., The determination of gas density - part 1: State of the art in the Nordic countries. Espoo 1997. Nordtest, NT Techn Report 353. 19p. NT Project No. 1254-95. 354 Tambo, M. & Søgaard, T., The determination of gas density - part 2: Intercomparison - Calibration of gas density meters with nitrogen. Espoo 1997. Nordtest, NT Techn Report 354. 91 p. 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